Holographic Theory, and Emergent Gravity As an Entropic Force

Home , Entropic force, Entropic gravity, Erik Verlinde

Holographic theory, and emergent gravity as an entropic force

Liu Tao December 9, 2020

Abstract In this report, we are going to study the holographic principle, and the possibility of describing gravity as an entropic force, independent of the details of the microscopic dynamics. We will ﬁrst study some of the deep connections between Gravity laws and thermodynamics, which motivate us to consider gravity as an entropic force. Lastly, we will investigate some theoretical criticisms and observational evidences on emergent gravity.

1 Contents

1 Introduction3

2 Black holes thermodynamics and the holographic principle3 2.1 Hawking temperature, entropy...... 3 2.1.1 Uniformly accelerated observer and the Unruh eﬀect...... 3 2.1.2 Hawking temperature and black hole entropy...... 6 2.2 Holographic principle...... 7

3 Connection between gravity and thermodynamics8 3.1 Motivations (from ideal gas and GR near event horizon)...... 8 3.2 Derive Einstein’s law of gravity from the entropy formula and thermodynamics 10 3.2.1 the Raychaudhuri equation...... 10 3.2.2 Einstein’s equation from entropy formula and thermodynamics.... 11

4 Gravity as an entropic force 13 4.1 General philosophy...... 13 4.2 Entropic force in general...... 13 4.3 Newton’s law of gravity as an entropic force...... 14 4.4 Einstein’s law of gravity as an entropic force...... 15 4.5 New interpretations on inertia, acceleration and the equivalence principle.. 17

5 Theoretical and experimental criticism 18 5.1 Theoretical criticism...... 18 5.2 Cosmological observations...... 19 20section.6

2 1 Introduction

The holographic principle was inspired by black hole thermodynamics, which conjectures that the maximal entropy in any region scales with the radius squared, and not cubed as might be expected for any extensive quantities. We will start with a brief introduction on black hole thermodynamics in Section 2 Black holes thermodynamics and the holographic principle where we address the Hawking temperature and the black hole entropy formula, also known as the Bekenstein–Hawking formula. We then investigate some deep connections between Gravity laws and thermodynamics in Section 3 Connection between gravity and thermodynamics. We first consider two examples from ideal gas and Einstein’s field equations near the black hole event horizon as motivations [1,2]. Then we derive the Ein- stein’s law of gravity from the entropy formula and the law of thermodynamics in all its glory following [1]. Afterwards, in Section 4 Gravity as an entropic force we follow [3] and explain gravity as an entropic force generated by changes in the information associated with the positions of material bodies in both non-relativistic and relativistic settings. With this logic in mind we recover the Newton’s law of gravity in a simple non-relativistic setting and Einstein’s field equation when we generalize these results to the relativistic case. What’s more, when space is emergent even Newton’s law of inertia and and Einstein’s equivalence principle need to be reexplained. So we will give new interpretations on Newton’s law of inertia and the meaning of acceleration (both concepts in the original theories are related to spacetime-specific notions such as coordinates) in this new emergent space scenario where only the fundamental concepts such as entropy and temperature are essential. The equivalence principle can also be explained more naturally and unavoidably in a theory in which space is emergent through a holographic scenario since both gravity and acceleration are emergent phenomena. Last but not least, we summarize some of the theoretical criticism and observational evidences about emergent gravity theory of [3] in Section 5 Theoretical and experimental criticism. Despite all of the theoretical evidences presented in this report, there is a decreasing research interest into the emergent gravity theories within the community in recent years as far as I can tell. Maybe the last section can provide some clues. We will use the mostly pluses signature (−, +, +, +) throughout and units with kB = c = h¯ = 1.1

2 Black holes thermodynamics and the holographic principle

2.1 Hawking temperature, entropy 2.1.1 Uniformly accelerated observer and the Unruh eﬀect We are going to show in explicit details that for a uniformly accelerated observer [4], the ground state of an inertial observer is seen as a mixed state in thermodynamic equilibrium

1for the most part, since sometimes havingh ¯ in our result explicitly for example can emphasize the its quantum nature.

3 with a non-zero temperature bath, and the observed frequency spectrum is given by familiar Planck’s law with the temperature T = 1/β = a/(2π), where a is the constant acceleration. In the reference of a stationary observer, the four-acceleration of our uniformly accelerated observer is aα =x ¨α = (0, a), where |a| = a is constant. We can convert this condition into a covariant form α β 2 ηαβx¨ x¨ = a (1) we also have the normalization condition

α β ηαβx˙ x˙ = −1 (2)

Without lose generality, let’s assume that the trajectory is contained in the t–x plane, i.e. a = (a, 0, 0). Switching to the light-cone coordinates

u = t − x and v = t + x (3)

the line elements can be written as (suppress the transverse coordinates y and z)

ds2 = −dt2 + dx2 = −dudv (4)

The acceleration equation1 and the normalization equation2 can be written as

u¨v¨ = −a2 andu ˙v˙ = 1 (5)

The two equations in5 can be combined to eliminate one of the variable say u, we obtain v˙ = ±a (6) v˙ This is integrated to be A v(τ) = exp(aτ) + C (7) a and, usingu ˙ = 1/v˙ we got 1 u(τ) = − exp(−aτ) + D (8) Aa go back to the original Cartesian coordinates, we obtain2 1 1 t(τ) = sinh(aτ) and x(τ) = cosh(aτ) (9) a a where we have chosen the integration constants such that the initial conditions t(0) = 0 and x(0) = 1/a are satisﬁed. Now consider a monochromatic wave for the inertial observer

φ(k) ∝ exp(−iω(t − x)) (10)

2This result is also in one of our homework questions.

4 An accelerated observer does not see a monochromatic wave however, but a superposition of plane waves with varying frequencies. We can see this by inserting the trajectory9 into the monochromatic wave iω iω φ(τ) ∝ exp − [sinh(aτ) − cosh(aτ)] = exp exp(−aτ) (11) a a As the next step, we want to determine its power spectrum P (ν) = |φ(ν)|2 and relate it to the Plank’s formula. First let us calculate the wave 11 in the Fourier space φ(ν). According to the Fourier transform, we obtain Z ∞ Z ∞ iω φ(ν) = dτφ(τ)e−iντ = dτ exp exp(−aτ) e−iντ (12) −∞ −∞ a change variable to y = exp(−aτ), we obtain 1 Z ∞ φ(ν) = dyyiν/a−1ei(ω/a)y (13) a 0 we notice the above integral is similar to the Gamma function integral Z ∞ dttz−1e−bt = b−zΓ(z) = exp(−z ln b)Γ(z) (14) 0 except our integral 13 is not convergent. Here we added an infinitesimal positive real quantity ε to ensure the convergence. Combine equations 13 and 14, we obtain 1 φ(ν) = · exp (−iν/a ln(−iω/a + )) · Γ(iν/a) (15) a iω ω iπ where we also have limε→0 ln − a + ε = ln a − 2 sign(ω/a). Use this we obtain 1 ω −iν/a φ(ν) = Γ(iν/a)e−πν/(2a) (16) a a for positive ω/a. Thus, the power spectrum is 1 P (ν) = |φ(ν)|2 = e−πν/aΓ(iν/a)Γ(−iν/a) (17) a2 use Euler’s reflection formula −π πa Γ(iν/a)Γ(−iν/a) = iν ν = πν (18) a · sin(iπ a ) ν sinh a equation 17 now becomes π e−πν/a β 1 P (ν) = = (19) a2 (ν/a) sinh(πν/a) ν eβν − 1 with β = 2π/a. This is the Planck’s formula. Therefore, classically, a uniformly accelerated detector will measure a thermal Planck spectrum with temperature T = 1/β = a/(2π). This phenomenon is called Unruh effect [5].

5 This result can also be easily generalized to quantum ﬁeld theory. Quantum mechanically, in the canonical quantization formalism3, we can follow the same steps and consider instead any quantum ﬁelds with harmonic expansions similar to the classical monochromatic wave we considered above, but with integration over all available frequency modes. As a result, the vacuum expectation value of the number density operator of scalar particles detected by a uniformly accelerated observer for example is [4] 1 hn˜ i = (20) Ω exp(2πΩ/a) − 1 which is again, the black body radiation with the temperature given by T = 1/β = a/(2π). In special relativity, an observer moving with uniform proper acceleration through Minkowski spacetime is conveniently described with Rindler coordinates, which are related to the stan- dard (Cartesian) Minkowski coordinates by

x = ρ cosh(σ) (21) t = ρ sinh(σ)

The line element in Rindler coordinates is

ds2 = −ρ2dσ2 + dρ2 (22)

1 where ρ = a and σ = aτ for an uniformly accelerated observer, see equation9. In the next section where we derive the black hole entropy, we are going to massage the Schwarzschild metric outside a black hole into the above 22 form and interpret the resulting Unruh temperature of stationary observer at the spatial inﬁnity as the well-known Hawking temperature. Combine the Hawking temperature and the second law of thermodynamics we can ﬁnally derive the black hole entropy formula.

2.1.2 Hawking temperature and black hole entropy As brieﬂy outlined above, in order to calculate the black hole entropy, let’s start with the Schwarzschild metric [2,7]

dr2 ds2 = −f(r)dt2 + + r2 dθ2 + sin2 θdφ2 (23) f(r)

where f(r) ≡ 1 − 2GM/r. Let’s consider the region near (but outside) the horizon r = rs = r→rs 0 2GM, f(r) = f (rs)(r − rs). Introducing the proper distance ρ dr dρ = √ (24) f we can write f(r) = K2ρ2 + ··· (25)

3In the path integral formalism, we can arrive the result by utilizing the formalism similarity between quantum statistical mechanics and quantum ﬁeld theory, i.e. the generating function in quantum ﬁeld theory v.s. the partition function in statistical mechanics [2,6].

6 1 0 1 where K ≡ f (rs) = is the surface gravity. So near the horizon, the metric 23 can 2 4GN M be rewritten as

2 2 2 2 2 2 2 2 2 2 2 2 ds = −K ρ dt + dρ + rs dΩ2 = −ρ dη + dρ + rs dΩ2 (26) with η = Kt = t . This metric is similat to the Rindler metric 22 except for the last 2rs term that is not relevant for our purposes. An observer at r = const is equivalent to an 1 observer with ρ = const. Such an observer has a constant proper acceleration a = ρ , and 1/2 1 the acceleration seen by O∞ is redshifted to be a∞ = a(r)f (r) = ρ · Kρ = K. The local temperature at ∞ (TH ) can be calculated by the acceleration and Unruh eﬀect a 1 hc¯ 3 T = = = (27) H 2π 8πGM 8πGM which is the Hawking temperature of the black hole! With the black hole temperature, it is straightforward to calculate the black hole entropy. The change in entropy when a quantity of heat dQ is added is

dQ 8πGM 8πGMdM c3 2GM 2 dA dS = = 3 dQ = = 4πd 2 = 2 (28) T hc¯ hc¯ 4¯hG c 4 · `P

2 p 3 where A = 4πrs is the event horizon area, and `P = Gh/c¯ is the Planck length. This means A S = 2 (29) 4 · `P This is the Bekenstein–Hawking formula!

2.2 Holographic principle The entropy of a thermodynamics system is normally extensive and proportional to its volume. However, as shown in equation 28, the entropy of a black hole is proportional to its area. It is as if the entropy/information of a black hole were to reside completely on its surface. The holographic principle was exactly inspired by this remarkable result [8,9], which conjectures that the maximal entropy in any region scales with the radius squared, and not cubed as might be expected. We are going to show this result in the following with a thought experiment. Consider [7,10] an isolated system mass E and entropy S0 in an asymptotic ﬂat spacetime. Let A be the area of the smallest sphere hat encompasses the system, and MA to be the mass of a black hole with the same horizon area. We must have E < MA, otherwise the system would be already a black hole. Now add MA −E energy to the system while keeping the area ﬁxed, we shall obtain a black hole with mass MA. Since the second law of thermodynamics, we have 0 SBH ≤ S0 + S (30) where S0 is the entropy of the added entropy. We then conclude A S0 ≤ SBH = 2 (31) 4`P

7 A That is to say the maximum entropy inside a region bounded by area A is 2 . 4`P This completes our derivation of the black hole entropy formula and the holographic principle. It’s the aim of this report to re-interpret Newton and Einstein’s laws of gravity in a non-relativistic and relativistic setting respectively with some basic assumptions such as the holographic principle and emergent spacetime. However before we jump in right away, in order to motivate our study on the emergent gravity scenario, let us investigate some deep connections between Gravity laws and the laws of thermodynamics ﬁrst. In particular, we are going to re-derive the Einstein’s law of gravity as an equation of state assuming the holographic principle and thermodynamics in the next section.

3 Connection between gravity and thermodynamics

3.1 Motivations (from ideal gas and GR near event horizon) There is a deep connection between Gravity laws and the laws of thermodynamics. It has been shown in [1,2] that one can derive Einstein’s field equation as an equation of state by combining the Bekenstein–Hawking formula 28 and the fundamental relation δQ = T dS relating heat, entropy, and temperature in thermodynamics. To see logically how this is possible, let us consider a similar example in the context of ideal gas [1]. Our goal is to derive the “field equation” of any thermodynamic systems given its entropy formula S(E,V ) and the thermodynamics relation dE = T dS − P dV . In general we have 1 ∂S P ∂S = and = (32) T ∂E V T ∂V and in the case of ideal gas, the entropy is given by the logarithm of the number of possible states, i.e. it goes like S = N log V + f(E) with some function f(E). The above thermodynamics relation 32 then yield PV = NT , which is the well-known equation of state of an ideal gas. Viewed in this way, if we can derive Einstein’s field equation from the entropy formula (holographic principle) and general rules of thermodynamics, the Einstein equation is de- moted to the status of an equation of state just like the ideal gas example. This perspective suggests that it may be no more appropriate to canonically quantize the Einstein equation than it would be to quantize the wave equation for sound in air [1], and one of the apparent difficulties in quantizing gravity than quantizing the ideal gas for example, I guess, is the non-linearly in the Einstein’s equation. In deriving the Einstein’s equation as an equation of state, the key idea is to demand the thermodynamic relation δQ = T dS holds for all local Rindler causal horizons through each spacetime point with δQ and T interpreted as the energy flux and Unruh temperature seen by an accelerated observer just inside the horizon. This requires that gravitational lensing by matter energy distorts the causal structure of spacetime in just such a way that the Einstein equation holds. Let’s see how this is done intuitively. Let’s consider an infinitesimal amount of heat δQ going through a local Rindler causal horizon through an arbitrary spacetime point. δQ is given by integrating the energy flux,

8 which depends on the energy-momentum tensor Tµν. On the other hand, the right hand side of the relation δQ = T dS is related to the area change δA through the Bekenstein–Hawking formula, which itself is determined by the curvature of spacetime Rµν through the Ray- chaudhuri equation. So as a result of this procedure, we have a equation relating the energy momentum tensor and the Ricci tensor. This relation turns out to be Einstein’s ﬁeld equation, as we will show in the following. Before we show the derivation explicitly, let’s look at yet another interesting case where you can clearly see the deep intricate relation between Gravity and Thermodynamics provided by Prof. Padmanabhan [2]. Let’s consider a static, spherically symmetric horizon, in a spacetime described by a metric dr2 ds2 = −f(r)dt2 + + r2 dθ2 + sin2 θdφ2 (33) f(r) notice unlike equation 23, this doesn’t have to be Schwarzschild metric. We can follow the same steps as section 2.1.2 and show that the metric reduces to the Rindler metric near the 1 0 horizon in the r − t plane with the surface gravity K = 2 f (a) and Unruh temperature a f 0(a) T = = (34) H 2π 4π where f(a) = 0 and f 0(a) 6= 0. Given the metric 33, we can write down the Einstein equation following the normal procedure. We obtain (1 − f) − rf 0(r) = − (8πG) P r2 (35) r with P = Tr is the radial pressure (spherically symmetric). Evaluate on the horizon r = a and use the fact that on the horizon f(a) = 0, we get the result

1 1 f 0(a)a − /G = 4πP a2 (36) 2 2

multiply the above equation by da on both sides and integrate we get

hcf¯ 0(a) c3 1 1 c4da 4π d 4πa2 − = P d a3 (37) 4π Gh¯ 4 2 G 3

where we have restored all theh ¯ and c. This is exactly the thermodynamics relation T dS − dE = P dV if we recognize

3 2 4 c 2 1 AH 2 ac 2 c S = 4πa = 2 ; E = Mc = c = a (38) 4Gh¯ 4 `P 2G 2G

p 3 where AH is the horizon area, and `P = Gh/c¯ is the Planck length as shown in the Bekenstein–Hawking formula above 28. The result shows that the Einstein’s ﬁeld equations evaluated on the horizon reduces to a thermodynamic identity. Some people believe this is not a simple reversing of the arguments or a coincidence, but due to a much deeper connection between Gravity and Thermodynamics.

9 3.2 Derive Einstein’s law of gravity from the entropy formula and thermodynamics Okay now let’s shut up and calculate. We are going to re-derive the Einstein’s ﬁeld equation relating the energy momentum tensor to the Ricci tensor from the Bekenstein–Hawking’s entropy formula and the law of thermodynamics. As shown in the intuitive discussion above, we are going to need the Raychaudhuri equation relating the amount by which light rays converge or diverge with the spacetime curvature. So ﬁrst of all, let’s derive the Raychaudhuri equation from the geodesic equation.4

3.2.1 the Raychaudhuri equation Let’s give a derivation of the Raychaudhuri equation we are going to use. Consider [6] a bundle of time-like geodesics xµ (τ, σ1, σ2, σ3) labeled by three real value σ1,2,3. Pick a point µ dxµ P on one speciﬁc geodesic and the tangent vector is V = dτ . We also have the geodesic µ ν µ dxµ equation V DµV = 0. Now consider the three deviation vectors W = dσ , then DW µ = V νD W µ = W νD V µ ≡ BµW ν (39) Dτ ν ν ν

µ where the last equality serve only as the deﬁnition of Bν , which tell us the rate of change of W. Also, the second equality comes from the fact that the Lie derivative of the deviation vectors W µ vanishes dW µ dV µ d2xµ d2xµ L W µ = V v∂ W µ − W ν∂ V µ = V vD W µ − W νD V µ = − = − = 0 V ν ν ν ν dτ dσ dσdτ dτdσ (40) v µ ν µ it follows that V DνW = W DνV , which is the second equality in equation 39. The idea is to compute the rate of change for Bµν. First of all, we can decomposed a generic 2-index tensor into three irreducible parts 1 B = σ + θP + ω (41) µν µν 3 µν µν

µν µ ν where Pµν ≡ g + V V is the projection operator in the three dimensional subspace 1,2,3 µν µν λ µλ spanned by σ . To see this, we can easily check P Vν = 0,P Pν = P . The trace- 1 1 less symmetric part σµν ≡ 2 (Bµν + Bνµ) − 3 θPµν describes shear, the antisymmetric part 1 µν µ ωµν = 2 (Bµν − Bνµ) describes rotation, and the trace part θ = P Bµν = DµV describes expansion. Now let’s diﬀerentiate Bµν, we get DB µν = V λD B = V λD D V = V λD D V + V λ [D ,D ] V Dτ λ µν λ ν µ ν λ µ λ ν µ λ λ λ σ (42) = Dν V DλVµ − DνV (DλVµ) − V RµλνVσ λ σ λ = −BµλBν − RσµλνV V

4I did not know this material will be discussed in the lectures while I was working on this draft. The following derivation found in A. Zee’s book [6] is quite similar to the lecture note.

10 where we have used the product rule in the second to last equality, and in the last equality the first term is zero because of the geodesic equation. So we can see, indeed, in the equation governing Bµν, the Riemann curvature tensor appears. This equation, known as the Raychaudhuri equation, tells us how Bµν varies as we move along a geodesic. In the next section where we derive the Einstein’s equation from the entropy formula and the law of thermodynamics, we are interested in how the expansion parameter θ changes, i.e. Dθ µν DBµν how a bundle of geodesics converges (or diverges). Since Dτ = g Dτ , contrast equation 42 µν Dθ with g we can get an equation for Dτ . The first term on the right hand side becomes µν λ νµ 1 µν 1 µν µν g BµλBν = BµνB = σµν + θPµν + ωµν σ + θP − ω 3 3 (43) 1 = σ σµν + θ2 − ω ωµν µν 3 µν We thus obtain the desired result relating the expansion parameter θ and the Ricci tensor Dθ 1 = − θ2 − σ σµν + ω ωµν − R V µV ν (44) Dτ 3 µν µν µν We are interested in non-rotational geodesics in what follows, so we have ωµν = 0. We finally obtain the equation for the expansion for time-like geodesics dθ 1 = − θ2 − σ2 − R V µV ν (45) dλ 3 µν For null geodesics [11], however, the expansion equation is slightly different. First of all, we need to define the null tangent vector with an affine parameter λ with dxa = a a a k dλ with k ka = 0 and k ξa = 0 (with ξ being the deviation vectors). We need to change µν the projection operator since now P kν 6= 0 is no longer true. Instead, we need to introduce ν ν an auxiliary null vector N with kνN = −1. Then the projection operator now can be µ µ chosen to be Pµν = gµν +kµNν +kνNµ. This satisfies k Pµν = 0 and N hµν = 0. This means that Pµν = 0 construsted this way will be two dimensional instead of three dimensional as we have in the time-like geodesics case above. The expansion equation for the case of null geodesics is dθ 1 = − θ2 − σ2 − R kakb (46) dλ 2 ab And this is the equation we are going to utilize in the next section.

3.2.2 Einstein’s equation from entropy formula and thermodynamics First of all, let’s specify our system [1]. The equivalence principle is invoked to view a small neighborhood of each spacetime point p as a piece of flat spacetime. It is always possible to choose a small spacelike 2-surface element P through p so that the expansion and shear vanish in a first order neighborhood of p. Therefore the θ2 and σ2 terms in the expansion equation 46 are higher order contributions that thus can be neglected compared with the last term when integrating to find the expansion θ near P. This integration yields

a b θ = −λRabk k (47)

11 for small enough λ. As illustrate above, for the small spacelike 2-surface element P one has an approximately flat region of spacetime with the usual Poincare’s symmetries according to the equivalence principle. Also, there is an approximate Killing vector field χa generating boosts orthogonal to P and vanishing at P. What’s more, according to the Unruh effect 27, the boost observer κ will experience a thermal state with temperature T = 2π , where κ is the acceleration of the Killing orbit on which the norm of χa is unity. The heat flow then is defined by the a boost-energy current of the matter, Ta,bχ , where Ta,b is the energy momentum tensor. Now consider any local Rindler horizon through a spacetime point p with Killing vector χa generating this horizon. The heat flux is given by the following integral Z a b δQ = Tabχ dΣ (48) H

In terms of the tangent vector ka, the Killing vector and the area element can be written as

χa = −κλka and dΣa = kadλdA (49)

where κ is the acceleration. Assuming the Bekenstein–Hawking relation 28, the entropy change the heat exchange can be written as dS = δA/4`2 , where δA is the area variation. The area variation δA can P R be further written in terms of the expansion parameter θ as δA = H θdλdA. The entropy change then is R R a b H θdλdA H λRabk k dλdA dS = 2 = − 2 (50) 4`P 4`P where we have used equation 47 in the last equality. Now we have equations for the heat ﬂux δQ and entropy change dS in equations 48 and 50. We can simply relate them through the law of thermodynamics δQ = T dS. Notice 2 that `P = G in natural unit, we obtain 1 T kakb = R kakb (51) ab 8πG ab

1 for any null-like ka. This implies Tab = 8πG Rab + fgab for some f. Local conservation of the original energy momentum tensor implies that Tab is divergence free and therefore this posts a constraint on the additional function f. Using the contracted Bianchi identity we obtain f = −R/2 + Λ for some constant Λ. We thus obtain the Einstein equation with the mysterious cosmological constant term 1 R − Rg + Λg = 8πGT (52) ab 2 ab ab ab We thus see if we assume the equilibrium thermodynamic relation δQ = T dS holds, where δQ is interpreted as the energy ﬂux through the local Rindler horizons and dS is propotional to the area change, the Einstein equation holds. It seems like the physics of gravity has to be arranged in precisely such a way as the form of Einstein’s equation so that the formula S ∝ A (instead of say S ∝ V ) holds.

12 4 Gravity as an entropic force

4.1 General philosophy As we have seen in the previous sections, there exists a deep and mysterious connection between the field equations of gravity and the horizon thermodynamics [1,2]. In the following we will argue that the central notion needed to derive gravity is information. More precisely, it is the amount of information associated with matter and its location, in whatever form the microscopic theory likes to have it, measured in terms of entropy. Changes in this entropy when matter is displaced leads to an entropic force [3], which as we will show takes the form of gravity. In this way, gravity should be understood from general principles that are independent of the specific details of the underlying microscopic theory, which coincides with the universality of gravity. In the following, we are going to start from first principles, using only spacetime-independent concepts like energy, entropy and temperature to show that Newton and Einstein’s laws of gravity appear naturally and practically unavoidably. In order to interpret gravity as an entropic force caused by a change in the amount of information associated with the positions of bodies of matter, we are going to use the holographic principle and assume that the amount of information associated with a given spatial volume is proportional to the area. On the other hand, the energy, that is equivalent to the matter, is distributed evenly over the degrees of freedom, and thus leads to a temperature. We then derive the entropic force associated with this process, which turns out to be the Newton’s law of gravity. We are also going to extent this reasoning in a relativistic setting to derive the Einstein’s equation.

4.2 Entropic force in general Before we derive Newton and Einstein’s Gravity laws as entropic forces, let’s first look at entropic force in general and see how it arises in thermodynamics. An entropic force acting in a system is an emergent phenomenon resulting from the entire system’s statistical tendency to increase its entropy, rather than from a particular underlying force on the atomic scale. In particular, unlike the fundamental interactions such as electromagnetic and weak forces, there is no fundamental field associated with an entropic force. For definiteness [3], let’s consider a system with entropy equals to S(E, x) = kB log Ω(E, x), where Ω(E, x) is the number of microscopic states. In the canonical ensemble the force F is introduced in the partition function Z Z(T,F ) = dEdxΩ(E, x)e−(E+F x)/kB T (53) the force F required to keep the system at equilibrium at state x, E is ∂S F = T (54) ∂x We can see from the above equation that entropic forces result from entropy gradient and always point in the direction of increasing entropy, and it is also proportional to the temperature of the system.

13 4.3 Newton’s law of gravity as an entropic force We now derive the Newton’s gravity law as an entropic force. First of all, from the holographic principle, the description of a volume of space can be thought of as N bits of binary information, encoded on a boundary to that region, a closed surface of area A. Each bit 2 of information requires an area equals to the Planck area `P, and the information is evenly distributed on the surface. Thus we get A Ac3 N = 2 = (55) `P hG¯ In order to derive Newton’s law of gravity, lets look at a flat non-relativistic space. Let’s consider a small piece of an holographic screen at the boundary, and a particle of mass m that approaches it from the side at which spacetime has already emerged. Let’s follow Bekenstein’s original thought experiments and postulate when a particle is one Compton wavelength from the holographic screen, it is considered to be part of the area. Let’s postulate that the change of entropy associated with the information on the boundary equals h¯ ∆S = 2πk when ∆x = (56) B mc or we can write it in a more general form and assume mc ∆S = 2πk ∆x (57) B h¯ This will be our basic assumption about the entropy change that is associated with the displacement of matter. Before we move on, let’s first give an motivation about the above postulate. First of all, the entropic force satisfies

F ∆x = T ∆S (58) combine this with our postulate 57 and use the Unruh temperature 27 here as T, we get F = ma, where a denotes the acceleration. So we have shown that starting from the postulate 57, we recover the Newton’s second law. Keep in mind that in the entropic gravity scenario we are considering, we will discard all the quantities related to spacetime (as it is emergent and not fundamental), such as coordinates and acceleration, while keepping quantities like entropy and temperature as more fundamental. As a result, we will use the Newton-equivalent entropy postulate 57 as our starting point from now on. Now we are going to derive the Newton’s gravity law from the postulate 57. First of all, combine with F ∆x = T ∆S, we get 2πk mc F = B · T (59) h¯ Let’s ﬁnd what T is. Since we are interested in thermal equilibrium, from the statistical equipartition theorem, the temperature of a system with N degrees of freedom and energy E (which in our case is equal to the mass of the material) is 2E 2Mc2 2Mc2`2 T = = = P (60) kBN kBN kBA

14 where we have used the formula for the number of degrees of freedom 55. Substitute 60 into 59 and use the fact that A = 4πr2, we get

2 ¯hG 2πkBmc 2Mc c3 GMm F = · 2 = 2 (61) h¯ kB4πr r We have recovered Newton’s law of gravitation! You may argue, as I said above, that the reason why we can recover the Newton’s law is simply that we reversed all of the usual arguments. This might be true, but the more important message here is the underlying logic, which states the origin of gravity: it is an entropic force. If true, this should have profound consequences, as we will discuss in the following sections.

4.4 Einstein’s law of gravity as an entropic force In order to derive Einstein’s law of gravity as an entropic force, let’s consider a static back- φ ground with a global time-like Killing vector ξa. We need a factor e representing the redshift factor that relates the local time coordinate to that at a reference point with φ = 0, where φ is the gravitational ﬁeld. A natural generation of the Newton’s potential in general relativity is [12] 1 φ = log (−ξaξ ) (62) 2 a The four velocity and acceleration can be written as 5

b −φ b b a b −2φ a b u = e ξ , a ≡ u ∇au = e ξ ∇aξ (63) we can rewrite the acceleration by using the Killing equation ∇aξb + ∇bξa = 0 and the expression for φ. We get 1 1 ab = e−2φξa∇bξ = e−2φ∇b (ξ ξa) = − e−2φ∇be2φ = −∇bφ (64) a 2 a 2 Similar to the non-relativistic case 57 in the previous section, we are going to follow Bekenstein and postulate that the change of entropy at the screen is 2πkB for a displacement by one Compton wavelength normal to the screen mc ∇ S = −2πk N (65) a B h¯ a

where Na is a unit outward pointing vector perpendicular to the screen S and to ξb. The extra minus sign compared to 57 comes from the fact that the entropy gradient is in the opposite direction as the outward vector (i.e. entropy increases when we cross from the outside to the inside). Same logic as the non-relativistic case, in order to give a motivation to postulate 65, we can look at the Unruh temperature and see that the resultant entropic force leads to Newton’s F = ma. More speciﬁcally, a natural generation of the Unruh temperature 1 T = eφN b∇ φ (66) 2π b 5 I do not not quite understand why we are ignoring the term proportional to ∇aφ in the acceleration. The original source I found is Equ. 11.2.3 in Wald’s book [12] where he talked about the concept of ”staying in place”.

15 where we have added the extra redshift factor eφ and the unit vector N b. Then the entropic force in this system is φ Fa = T ∇aS = −me ∇aφ (67) which is exactly F = ma 64 with an additional factor eφ due to the redshift. Now let’s derive Einstein’s field equation as an entropic force. First of all, we immediately see that we cannot apply the same logic as the non-relativistic case since now we want to find a field equation relating the energy momentum content and the curvature of spacetime instead of a law for the ‘force’. We have seen above the local temperature 66 and our postulate 65 are equivalent. Now let’s start with the holographic principle. We assume that holographic screen is enclosing a certain static mass configuration with total mass M. The bit density on the screen is then given by dAc3 dN = (68) Gh¯ Again by equipartition we have Z 2 1 Mc = kB T dN (69) 2 S substitute in equations 66 and 68 we get 1 Z M = eφ∇φ · dA (70) 4πG S which is precisely Komar’s definition of the mass contained inside an arbitrary volume inside any static curved space time. It can be derived by assuming the Einstein’s field equation. In our reasoning, however, we are coming from the other side. In the following I will follow chapter 11 in book [12] to derive Einstein’s field equation from Komar’s mass formula 70. Start with equation 70. Notice the integral is taken over any topological 2-sphere, which encloses all of the sources. Let’s use the equality 64 to write the right hand side integral as an integration over the Killing vector Z 1 −φ a b M = − e ξ ∇bξaN dA (71) 4πG S

Use Killing’s equation ∇aξb = ∇[aξb], we can write the above as Z kB a,b M = − N ∇[aξb]dA (72) 8πG S where N a,b = 2e−φξ[aN b] is the normal ‘bi-vector’ to S. We can continue to write the above equation to as an integration over a two-form Z 1 a b c d M = − dx ∧ dx abcd∇ ξ (73) 8πG S

where abcd is the volumn element on spacetime associated with the spacetime metric. We can convert the surface integral 73 to a volumn integral using Stokes’s theorem Z 3 c d M = − ∇[e ab]cd∇ ξ (74) 8πG Σ

16 In order to see what the integrand is, let’s calculate6 efab c d efab c d ∇f abcd∇ ξ = abcd∇f ∇ ξ [e f] = −4∇f ∇ ξ f e (75) = 4∇f ∇ ξ e f = −4Rf ξ

And then, multiply the above equation with elmn and contract over e we get 2 ∇ ∇cξd = Re ξf (76) [l mn]cd 3 f elmn Substitute this into equation 74 we get Z 3 c d M = − ∇[e ab]cd∇ ξ 8πG Σ Z 1 c d = − ∇[e ab]cd∇ ξ 4πG Σ Z (77) 1 d f = − Rf ξ deab 4πG Σ Z 1 a b = Rabn ξ dV 4πG Σ a d where n is the normal vector to Σ, so that εabc = n dabc is the natural volume element represented by dV . On the other hand, we can see that the energy momentum tensor emerges naturally on the left hand side of the original equality 70 to rewrite the total mass. So we end up with Z Z 1 a b 1 a b 2 Tab − T gab n ξ dV = Rabn ξ dV (78) Σ 2 4πG Σ where the particular combination of the stress energy tensor on the left hand side can be ﬁxed by combining the conservation laws of the tensors that occur in the integrals 7. Notice that equation 78 only gives us only a certain component of the Einstein’s equation. To get to the full Einstein we now use a similar reasoning as Jacobson [1] for time-like screens instead of null-like screens. We will end up with the full Einstein equations, same as what we had in equation 52 1 R − Rg + Λg = 8πGT (79) ab 2 ab ab ab 4.5 New interpretations on inertia, acceleration and the equivalence principle The acceleration due to the entropic force (gravity), can be calculated with F ∆S 2c2 ∆S a = = T · ∆x = (80) m m kBn ∆x 6We used one of the results of the homework problems. 7This particular choice of the left hand side, in my opinion, is similar to what we did in Equ.[51] where we applied additional constraints/arguments such as the contracted Bianchi identity that we assume holds

17 where n denotes the amount of new bits of information created by m when m ﬁnally merges with the microscopic degrees of freedom on the screen. Equation 80 shows that the acceleration is proportional to the entropy gradient 8. This will be one of our main principles: inertia is a consequence of the fact that a particle in rest will stay in rest because there are no entropy gradients. This new way of thinking can also shed light on the equivalence principle. The equivalence principle tells us that redshifts can be interpreted in the emergent space time as either due to a gravitational ﬁeld or due to the fact that one considers an accelerated frame. However, as shown in the last subsection, both views are equivalent in the relativistic setting, but neither view is microscopic or fundamental in the sense that they are both emergent phenomena.

5 Theoretical and experimental criticism

5.1 Theoretical criticism In its current form, entropic gravity has been severely challenged on formal grounds. Let’s look at two examples. Matt Visser points out [13] that if we wish to retain key parts of Verlinde’s proposal (an Unruh-like temperature, and entropy related to “holographic screens”) and to model conservative forces in the general Newtonian case (i.e. for arbitrary potentials and an unlimited number of discrete masses), the resulting model with a single heat bath is at best orthogonal to Verlinde’s proposal. So that leads to unphysical requirements for the required entropy and involves an unnatural number of temperature baths of differing temperatures. Visser concludes that based on the work of Jacobson [1], Padmanabhan [2], and others, there are good reasons to suspect a thermodynamic interpretation of the fully relativistic Einstein equations might be possible. Whether the specific proposals of Verlinde [3] are anywhere near as fundamental is yet to be seen — the rather baroque construction needed to accurately reproduce n-body Newtonian gravity in a Verlinde-like setting certainly gives one pause. Tower Wang also considered [14] a wide class of modified entropic gravity models of the form aAc2 E = f(a, A) (81) 4πG where f(a, A) is a model-dependent function of acceleration and area of the holographic surface 9. Notice in the case of Verlinde [3], f = 1. Wang has shown that the inclusion of the covariant conservation law of energy-momentum tensor severely constrains viable modifications of entropic gravity. In addition, he has shown that assuming the FLRW metric (i.e. assume cosmological homogeneity and isotropy), there

8We are considering a process where a particle of mass m is approaching the holographic screen and we are trying to re-interpret its acceleration. So I guess even though the background (M at the center for example) is in thermal equilibrium there is still entropy gradient caused by the displacement ∆x of m relative to M. 9My best guess for how to choose the area A of mass m for this generalized entropic gravity model would be the horizon area of the corresponding black hole with the same mass m, which is bounded by the minimum encompassing area of the particle.

18 is a discrepancy in deriving cosmological equations unless a condition for f(a, A) is satisfied 10 ¨ 2 Rr − ∂ f + ∂t∂rf = 0 (82) t R˙ where R(t) is the cosmological scale factor. For a model with a specific function of f, the condition is not always satisfied. So Wang concluded that the modified entropic gravity models of form 81, if not killed11, should live in a very narrow room to assure the energy- momentum conservation and to accommodate a homogeneous isotropic universe.

5.2 Cosmological observations Verlinde proposed [15] that the observed excess gravity in galaxies and clusters that is normally accounted for by the mysterious ‘dark matter’ is a consequence of entropic gravity. He shows that the emergent laws of gravity contain an additional ‘dark’ gravitational force describing the ‘elastic’ response due to the entropy displacement. Verlinde gives an estimate of the excess gravity in terms of the baryonic mass, Newton’s constant and the Hubble acceleration scale a0 = cH0, and argues that this additional ‘dark gravity force’ explains the observed phenomena in galaxies and clusters currently attributed to dark matter. A team from Leiden Observatory [16] observed statistically the lensing effect of gravitational fields at large distances from the centers of 33,613 galaxies, found that those gravitational fields were in good agreement with the prediction from entropic gravity. Whereas using conventional gravitational theory, the lensing effect of gravitational fields implied by these observations (as well as from measured galaxy rotation curves for example) could only be ascribed to a particular distribution of dark matter. However, there are also other cosmological observations that do not agree with the entropic gravity predictions. For example, Nieuwenhuizen [17] focuses on the good lensing and X-ray data for A1689, and finds out the need for dark matter starts near the cluster centre, where Newton’s law is still supposed to be valid for entropic gravity. Nieuwenhuizen concludes that entropic gravity is strongly ruled out unless additional (dark) matter like eV neutrinos is added. Also, Kris Pardo [18] developed equations for entropic gravity’s velocity curve predictions given a realistic, extended mass distribution, and applied it to isolated dwarf galaxies. He found that entropic gravity severely underpredicts the maximum velocities for those galaxies with measured velocities v > 165km/s. Kris concluded that rotation curves of these isolated, HI gas-rich, nearly spherical dwarf galaxies would provide the definitive test of entropic gravity.

10This means that the generalized entropic gravity model 81, according to Wang [14], is severely limited if not completely killed. 11Verlinde’s idea with f being a constant 1 agrees with Equ. 82 however. So I guess Wang’s objection is against a even more generic (but similar) emergent gravity model than Verlinde’s.

19 6 Final thoughts on this12

Despite all the theoretical and observational evidences against the emergent gravity scenario, there are still some aesthetically pleasing aspects in its formalism. Just to name a few: the deep connections between Gravity and Thermodynamics in general and the derivation of Newton’s and Einstein’s laws of gravity with spacetime-independent quantities such as energy, entropy and temperature by assuming linearity 13 between the entropy change associated with the position change of the material bodies and the matter displacement are all quite shocking to me the first I saw it. Some people might argue these are simply due to a reserve of arguments, and practically speaking both directions of the arguments are equivalent since they obtain the same equations of motion. This might be true but in my opinion, this does not necessarily mean the other side of arguments is unnecessary or simply redundant. Let me briefly illustrate what I mean here. For any fundamental physical (or even mathematical for that matter) theories, we start with some basic assumptions as our starting points and then use mathematical logic to derive and present its consequences as mathematical equations of motion mostly. In my opinion, however, a complete understanding of a physical theory is not just about the equations, but more about the fundamental assumptions that lead to the equations and reflect our understanding of the physical process. In this sense, even though we arrived at the same equations in the emergent gravity scenario as in the traditional approaches, they still provide different insights about our understanding of gravity, i.e. it is an emergent phenomenon with no fundamental field associated with it. The only way to distinguish them (the fundamental assumptions or the way we understand the gravity) from each other is of course by observations. All of the observational evidences against emergent gravity that I can find however, including the ones I listed above, are all suggesting emergent gravity is not enough by its own in explaining the observed excess gravity in galaxies and clusters. But that does not completely rule out emergent gravity as a useful interpretation of gravity. I am happy to see more decisive evidences in the future, if any.

12This section summarizes some thoughts I had while preparing this draft. It is not mature in anyway and shouldn’t considered as part of the review of this topic. 13See assumptions 57 and 65. It would be better if these assumptions can arise from a deeper argument instead of the simple agreements with the known physical laws, as provided in this report. But again, we can treat the linearly as the ﬁrst principle of our emergent gravity theory.

20 References

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